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# Galois Representations with Non-Surjective Traces

Published online by Cambridge University Press:  20 November 2018

## Abstract

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Let $E$ be an elliptic curve over $\mathbb{Q}$ , and let $r$ be an integer. According to the Lang-Trotter conjecture, the number of primes $p$ such that ${{a}_{p}}\left( E \right)=r$ is either finite, or is asymptotic to ${{C}_{E,r}}\sqrt{x}/\log x$ where ${{C}_{E,r}}$ is a non-zero constant. A typical example of the former is the case of rational $\ell$ -torsion, where ${{a}_{p}}\left( E \right)=r$ is impossible if $r\equiv 1\,\left( \bmod \,\ell \right)$ . We prove in this paper that, when $E$ has a rational $\ell$ -isogeny and $\ell \ne 11$ , the number of primes $p$ such that ${{a}_{p}}\left( E \right)\equiv r\,\left( \bmod \,\ell \right)$ is finite (for some $r$ modulo $\ell$ ) if and only if $E$ has rational $\ell$ -torsion over the cyclotomic field $\mathbb{Q}\left( {{\zeta }_{\ell }} \right)$ . The case $\ell =11$ is special, and is also treated in the paper. We also classify all those occurences.

## Keywords

Type
Research Article
Information
Canadian Journal of Mathematics , 01 October 1999 , pp. 936 - 951

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